New Identities of Fractional S-Transform with Its Applications

Abstract

The current work provides a comprehensive and integrated introduction to the principles, properties and applications of the S-transform (ST) and fractional S-transform (FrST). The ST, which is a significant tool in signal processing, is a conceptual version of the FT with a Gaussian window function. It has been observed from the literature study that only linearity, scaling, timeshifting and convolution theorem of ST were documented. This led to the findings of remaining properties of ST in order to establish it as a complete transform technique. Along with this, a new better definition of convolution theorem for ST has also been presented. The FrST is a generalisation of the classical ST. The FrST has demonstrated to be a valuable technique for an analysis of a non-stationary signals. The FrST also acts as a time-frequency representation method with the frequency dependent resolution. Some of the remaining properties of FrST are proposed in this work so as to establish it as a complete transform technique. The proposed properties are convolution theorem, Parseval’s theorem, correlation theorem and sampling propositions. It will provide an appropriate and reasonable model for sampling and restoration of the signal for real uses. Moreover, two kinds of reconstruction error, namely truncation error and aliasing error arises due to sampling were also discussed.Multiresolution analysis (MRA) has recently become important, and even essential, in signal analysis and image processing. As one of the famous family members of the MRA, the wavelet transform (WT) demonstrated itself in numerous successful applications in various fields, and become one of the utmost powerful tools in the fields of signal analysis and image processing. Due to the fact that only the scale info is supplied in WT, the applications with the help of WT may be restricted when the totally referenced frequency and phase information are required. The FrST is a proposed multiresolution transform that supplies the fully referenced frequency and phase information. In the areas where ST and FrFT are used, the performance can be enhanced through the use of FrST. In addition, it has a close relationship with other transforms like Fourier transform (FT), and WT. To expand the applicability of FrST as a mathematical transform tool, MRA is used. Finally, the applications of proposed convolution theorem are demonstrated on multiplicative filtering (MF) for electrocardiogram (ECG) signal and linear frequency modulated (LFM) signal under AWGN channel. The FrST can be applied for other applications of non-stationary signal analysis, radar signal processing and also in image processing.